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%I #130 Apr 07 2023 18:32:06
%S 0,1,2,6,12,25,48,91,168,306,550,979,1728,3029,5278,9150,15792,27149,
%T 46512,79439,135300,229866,389642,659111,1112832,1875625,3156218,
%U 5303286,8898708,14912641,24961200,41734339,69705888,116311074,193898158,322961275,537492672
%N a(n) = n*Fibonacci(n).
%C Number of levels in all compositions of n+1 with only 1's and 2's.
%C Apart from first term: row sums of the triangle in A131410. - _Reinhard Zumkeller_, Oct 07 2012
%C Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third one. - _Sergey Kitaev_, Dec 08 2020
%D Jean Paul Van Bendegem, The Heterogeneity of Mathematical Research, a chapter in Perspectives on Interrogative Models of Inquiry, Volume 8 of the series Logic, Argumentation & Reasoning pp 73-94, Springer 2015. See Section 2.1.
%H Reinhard Zumkeller, <a href="/A045925/b045925.txt">Table of n, a(n) for n = 0..1000</a>
%H Russell Euler, <a href="https://fq.math.ca/Scanned/28-3/elementary28-3.pdf">Problem B-670</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 28, No. 3 (1990), p. 277; <a href="https://www.fq.math.ca/Scanned/29-3/elementary29-3.pdf">Application of Generating Functions</a>, Solution to Problem B-670 by Russell Jay Hendel, ibid., Vol. 29, No. 3 (1991), p. 278.
%H Rigoberto Flórez, Robinson Higuita and Alexander Ramírez, <a href="https://arxiv.org/abs/1808.01264">The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials</a>, arXiv:1808.01264 [math.NT], 2018.
%H Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics, Vol. 26, No. 3 (2019), Article P3.26.
%H Silvia Heubach and Toufik Mansour, <a href="https://arxiv.org/abs/math/0310197">Counting rises, levels and drops in compositions</a>, arXiv:math/0310197 [math.CO], 2003.
%H Kai Ting Keshia Yap, David Wehlau and Imed Zaguia, <a href="https://arxiv.org/abs/2101.12061">Permutations Avoiding Certain Partially-ordered Patterns</a>, arXiv:2101.12061 [math.CO], 2021.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F G.f.: x*(1+x^2)/(1-x-x^2)^2.
%F G.f.: Sum_{n>=1} phi(n)*Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} n*Fibonacci(n)*x^n, where phi(n) = A000010(n) and Lucas(n) = A000204(n). - _Paul D. Hanna_, Jan 12 2012
%F a(n) = a(n-1) + a(n-2) + L(n-1). - _Gary Detlefs_, Dec 29 2012
%F a(n) = F(n+1) + Sum_{k=1..n-2} F(k)*L(n-k), F = A000045 and L = A000032. - _Gary Detlefs_, Dec 29 2012
%F a(n) = F(2*n)/Sum_{k=0..floor(n/2)} binomial(n-k,k)/(n-k). - _Gary Detlefs_, Jan 19 2013
%F a(n) = A014965(n) * A104714(n). - _Michel Marcus_, Oct 24 2013
%F a(n) = 3*A001629(n+1) - A001629(n+2) + A000045(n-1). - _Ralf Stephan_, Apr 26 2014
%F a(n) = 2*n*(F(n-2) + floor(F(n-3)/2)) + (n^3 mod 3*n), F = A000045. - _Gary Detlefs_, Jun 06 2014
%F E.g.f.: x*(exp(-x/phi)/phi + exp(x*phi)*phi)/sqrt(5), where phi = (1+sqrt(5))/2. - _Vladimir Reshetnikov_, Oct 28 2015
%F This is a divisibility sequence and is generated by x^4 - 2*x^3 - x^2 + 2*x + 1. - _R. K. Guy_, Nov 13 2015
%F a(n) = L'(n, 1), the first derivative of the n-th Lucas polynomial evaluated at 1. - _Andrés Ventas_, Nov 12 2021
%F Sum_{n>=0} a(n)/2^n = 10 (Euler, 1990). - _Amiram Eldar_, Jan 22 2022
%p a:= n-> n*(<<0|1>, <1|1>>^n)[1,2]:
%p seq(a(n), n=0..37); # _Alois P. Heinz_, May 07 2021
%t Table[Fibonacci[n]*n, {n, 0, 33}] (* _Zerinvary Lajos_, Jul 09 2009] *)
%t LinearRecurrence[{2, 1, -2, -1}, {0, 1, 2, 6}, 34] (* or *)
%t CoefficientList[ Series[(x + x^3)/(-1 + x + x^2)^2, {x, 0, 35}], x] (* _Robert G. Wilson v_, Nov 14 2015 *)
%o (Magma) [n*Fibonacci(n): n in [0..60]]; // _Vincenzo Librandi_, Apr 23 2011
%o (PARI) Lucas(n)=fibonacci(n-1)+fibonacci(n+1)
%o a(n)=polcoeff(sum(m=1,n,eulerphi(m)*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n) \\ _Paul D. Hanna_, Jan 12 2012
%o (PARI) a(n)=n*fibonacci(n) \\ _Charles R Greathouse IV_, Jan 12 2012
%o (PARI) concat(0, Vec(x*(1+x^2)/(1-x-x^2)^2 + O(x^100))) \\ _Altug Alkan_, Oct 28 2015
%o (Haskell)
%o a045925 n = a045925_list !! (n-1)
%o a045925_list = zipWith (*) [0..] a000045_list
%o -- _Reinhard Zumkeller_, Oct 01 2012
%Y Partial sums: A014286. Cf. A000045.
%Y Cf. A099920, A023607.
%K nonn,easy
%O 0,3
%A _Jeff Burch_
%E Incorrect formula removed by _Gary Detlefs_, Oct 27 2011
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